Leibniz’s differential calculus applied to the catenary Olivier Keller, agrégé in mathematics, PhD (EHESS) Study of the catenary was a response to a challenge laid down by Jacques Bernoulli, and which was successfully met by Leibniz as well as by Jean Bernoulli and Huygens: to find the curve described by a piece of strung suspended from its two ends. Stimulated by the success of this initial research, Jean Bernoulli put forward and solved similar problems: the form taken by a horizontal blade immobilised on one side, with a weight attached to the other; the form taken by a piece of linen filled with liqueur; the curve of a sail. Challenges among scholars In the 17th century, it was customary for scholars to set each other challenges in alternate issues of journals. Leibniz, for example, challenged the Cartesians – as part of the controversy surrounding the laws of collision, known as the “querelle des forces vives” (vis viva controversy) – to find the curve down which a body falls at constant vertical velocity (isochrone curve). Put forward in the Nouvelle République des Lettres of September 1687, this problem received Huygens’ solution in October of the same year, while Leibniz’s appeared in 1689 in the journal he had created, the Acta Eruditorum. Another famous example is that of the brachistochrone, or the “curve of the fastest descent”, which, thanks to the new differential calculus, proved to be not a circle, as Galileo had believed, but a cycloid: the challenge had been set by Jean Bernoulli in the Acta Eruditorum in June 1696, and was resolved by Leibniz in May 1697. It was also in this journal, in May 1690, that Jacques Bernoulli suggested that Leibniz examine if his new calculus could solve the problem of the catenary. Leibniz did indeed solve this problem, though he did not publish the solution – to give other mathematicians time to try their hand at it – until 1690. Huygens, Jacques and Jean Bernoulli successfully solved the problem in the imparted time.1 1. As Marc Parmentier indicates in his annotations accompanying Leibniz’s text (Éditions Vrin): “The historical prestige of the question and the competition solemnly installed by Leibniz had put the catenary in everyone’s minds and in the years to come paved its way to the career of the new cycloid. However, the particularity of the 1 Figure 1: Brachistochrone curve. The arc of the cycloid AB (in red) is the curve through which a weight leaving from A will arrive at B in the shortest amount of time. It should be noted that some of the curves studied by Leibniz were already known, and that many of their properties had already been established through purely geometric means or pre-differential methods. The cycloid, for example, was one such case, but differential calculus could pride itself on having considerably simplified the proofs of the properties already established. In contrast, the catenary was a curve discovered thanks to the new calculus, which could therefore boast of having genuinely advanced the art of inventing, of which Leibniz so fond. Moreover, the author implies that he was the first to have noticed and used the link between this curve and logarithms, a link that makes it possible to “construct” the latter simply with a suspended piece of string. Indeed, I observed that the fecundity of this curve is equalled only by the ease by which it is realised, which places it ahead of all the Transcendents [transcendental curves]. In fact, we can obtain and trace it with little difficulty, through a physical type of construction, by hanging a piece of string or better still a catenary (of invariable length). And once, thanks to this, we have its outline, we can bring to light … all the Logarithms we may wish … We will untangle Leibniz’s difficult text by first giving the differential equation of the catenary, which, though it forms the basis of the text, is absent from it. From this, we will deduce point by point the construction of the catenary, and, inversely, the construction of logarithms from a “suspended piece of string”. Finally, we will deduce some of the properties of the catenary, stated by the author without demonstration. catenary, which no doubt contributed to its prestige, is that it is not a mechanical curve in Descartes’ sense, but a static curve. As such, it outclasses the cycloid in the scale of the transcendent curves to the extent that it cannot be constructed through a composition of movements.” 2 Figure 2: The catenary is the form taken by a piece of string, or a chain, suspended from its two ends at equal height. THE EQUATION OF THE CATENARY We can retrieve this equation thanks to a passage from Montucla:2 “We cannot, we believe, avoid putting geometrician readers some way down the path to the solution of this curious and difficult problem. To do so we will borrow the subtle analysis given by Jean Bernoulli in his Lectiones calculi integralis.” Figure 3: The equation of the catenary, established by Jean Bernoulli, stems from the similarity between the triangle CDH and the characteristic triangle (dx, dy, ds). Note the inversion of the axes compared with contemporary conventions. The constant a below is the abscissa of the point S (on the vertical axis). Jean Bernoulli’s reasoning is based on a property of statics, according to which the weight of section SC of the string (proportional to the length s of this 2. Histoire des mathématiques, Tome II, p. 468. 3 section) is to the constant tension (noted a) of the string in S as CH is to DH, where CD is the tangent to the string in C (Fig. 2). By returning to Montucla’s (slightly modernised) text, we can establish this property as follows. The equilibrium of a portion SC of the string (Fig. 2) results from the equilibrium of three forces: the tension a of the string in S, the tension T of the string in C, and the weight s (by way of a certain choice of unit) of the portion of the string SC. The tensions are “powers” (forces) directed by the tangents to the points in question; the tension in S is therefore directed horizontally, and the tension in C is directed following CD.3 Taking the components of the three forces brought to point D, we therefore have: • horizontally: a and T× cos CDH, therefore a = T× cos CDH • vertically: the weight s and T× sin CDH, therefore s = T× sin CDH s CH We deduce that = tanCDH = a DH CH dx Considering the characteristic triangle in C, we have = . The DH dy s dx differential calculus of the catenary is therefore = , “an equation which, a dy adx treated adroitly”, Montucla tells us, “will be reduced to dy = ”. x22 - a This “adroit” treatment, which Montucla leaves up to the geometrician s2 dx 2 dx 2 reader, should look like this: as ds2 = dx 2 + dy 2 , == , a2 dy 2 ds 2 - dx 2 dx s therefore = and, consequently, by integrating, x = a22 + s , hence ds a22 + s s = x22 - a . By carrying this value of s to the initial equation , we obtain adx the equation sought: dy = (1) x22 - a 3. Importantly, the tension a in S is independent of the choice of point C. 4 Note that we obtain the equation4 of only part of the curve, the increasing part, because the coordinates are pairs of positive numbers. The difficulties that such a restriction might raise, at a time when negative numbers were not yet recognised as legitimate, can possibly be compensated here by symmetrical considerations: we have the same curve “on the other side”. CONSTRUCTION OF THE CATENARY Here is a geometric construction of the curve, without the aid of any string or any catenary, and without the supposition of quadratures, a construction which, to my mind, must be judged the most perfect that can be obtained for the transcendents, and the one most consistent with the analysis. The text that follows this passage is difficult for the contemporary reader, all the more so since Leibniz, devilishly elliptical as he often is – and who has not even given the equation of the catenary – does not explain the link between the latter and a “logarithmic curve”, which is nevertheless the essential link in his dy dx text. Here is a differentiation possible at this time.5 We have = , and we as xdx have seen that , therefore ds = . As a consequence:6 s dy dx ds d(x+s) = = = a s x x + s Hence the solution is y = alog(x+s) ; the constant is determined by setting down the condition y = 0 for x = a (Fig. 2), which gives: x + s y = alog( ) (because for y = o, we have s = length of the arc = 0) a We will take a = 1 as Leibniz suggests, hence ultimately: s = x22 - a y = log(x + s) (2) 2 xx y = a ln + - 1 aa 4. In contemporary notations, the solution of this equation is . The reader will be yy a - x = eaa + e able to verify that 2 , which is indeed the equation of a catenary, with Leibniz’s customary inversion of axes. 5. Following an idea by Dominique Bénard of the Mans Irem (Institut de recherche sur l’enseignement des mathématiques). 6. Here we are applying a property linked to fractions, if a/b = c/d then this relation is equal to (a+c)/(b+d). 5 We observed above that equation (1) concerns only half of the curve. The other half, which is symmetrical to the first in relation to the vertical axis, will have the following equation: 1 y=- log(x+s)=log =log(x-s) (3) x + s since (x - s)(x + s) = x2 - s 2 = a 2 = 1 .